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Home , Collision theory, Transition state theory

On Rate Constants: Simple , Arrhenius Behavior, and Theory

17th February 2010

0.1 Introduction Up to now, we haven’t said much regarding the rate constant k. It should be apparent from the discussions, however, that:

• k is constant at a specific , T and pressure, P

• thus, k = k(T,P) (rate constant is temperature and pressure depen- dent)

• bear in mind that the rate constant is independent of , as the , or velocity itself is treated explicity to be concen- tration dependent

In the following, we will consider how the physical, microscopic detaails of reactions can be reasoned to be embodied in the rate constant, k.

1 Simple Collision Theory

Let’s consider the following gas-phase : A + B → Products

The reaction rate is straightforwardly:

1 Rate = k[A]α[B]β = k[A]1[B]1 α = 1 β = 1 N N = k A B V = volume V V Recall previous discussions of the total collisional frequency for heteroge- neous reactions:

8kT ∗ ∗ ZAB = σAB nA nB s πµ ∗ NA ∗ NB where the nA = V , nB = V are number of molecules/particles per unit volume. We can see the following:

8kT NA NB ZAB = σAB s πµ V V

k Here we see that the concepts| of collisions{z } from simple kinetic theory can be fundamentally related to ideas of reactions, particularly when we consider that elementary reactions (only for which we can write rate expressions based on and order mapping) can be thought of proceeding due to collisions (or interactions of some sort) of monomers (unimolecular), dimers(bimolecular), trimers (trimolecular), etc.

If we are to naively say that reactions occur due only to collisions of parti- cles (keep in mind the nature of the system – gas-phase,elementary reaction), then we can at the zero’th order equate the maximum reaction rate to the total collisional frequency for heterogeneous pairs:

max R = ZAB

max 1 1 8kT NA NB k [A] [B] = σAB s πµ V V

k | {z } max 8kT k ≡ σAB s πµ

2 1.1 Simple Collision Theory: Caveats Simple collision theory (SCT), we see from above, remarkably predicts an expression for the microscopic rate constant that relates to the dimensions of the reactive species, their mass, and temperature. The form we determined above assumes two things: • All collisions are of sufficient energy that chemical transformation can occur

• Steric/orientational nature of collision is always correct/accommodating Thus, we need to consider the effects of collision energy and collision sterics and/or orientation in our discussion of simple collision theory and its application to defining the reate constant, k.

• For energetic considerations, we can empirically add a factor to ac- count for the probability of a collision having a sufficient energy, vis-a-vis, Emin, for collision. When we multiply the total collisional frequency by this probability, we can describe the fraction of collisions that will energetically be able to progress from reactants to products. The energy probability is take to be Boltzmann-like:

−Emin P robability(Emin) ∝ e RT

3 • For steric/orientational nature of collision, we introduce a steric factor (empirically), p • p < 1 generally Thus, we can write a more general expression for the collision theory based reaction rate as:

−Emin Rate = ZAB p e RT

E 8kT − min NA NB = p σAB e RT s πµ V V Thus we arrive at a corrected Simple collision theory expresion for the rate constant:

8kT −Emin RT kSCT = p σAB e s πµ

8kT Emin ln(kSCT ) = ln p σAB − " s πµ # RT Note the temperature dependence of the rate constant with SCT: 1 kSCT ∝ T 2 .

In general, we can write this expression for the rate constant as:

m −Emin/RT kgeneral = c T e (1)

2 Arrhenius Temperature Dependence

1 In general, experiments do not suggest a T 2 temperature dependence of the rate constant. Moreover, experiments demonstrate for most reactions that 1 the temperature dependence of ln(k) is linear with T . Thus, Arrhenius pro- posed the following relation between temperature and the rate constant:

−Eactivation kArr = A e RT (2)

4 Thus,

E 1 ln(k ) = ln(A) − activation Arr R T (3)

The energy is the (as discussed above) and the A value is a temperature-independent freqency factor, or pre-exponential factor.

Plotting ln(k) versus T −1 will yield a straight line with slope −Eactivation equal to R and y intercept of ln(A).

The pre-exponential factor in this case if independent of temperature, con- trasted with the SCT result from above. Though many reactions (across the spectrum of reaction orders, mechanisms, etc.) follow Arrhenius behavior, there are exceptions (as always).

3 Activated Complex Theory; The- ory

In this section, we will consider a further refinement of our formulation of a reaction on the atomic level. Up to now, we have not entertained the possibility of short-lived, highly unstable intermediates appear- ing/being generated upon the initial ”collision” of reactive species. Eyring and co-workers postulated the presence of these highly unstable, fleeting, transition states, and furthermore suggested an equilibrium between this transition state and the reactive species. The transition state is also consid- ered an activated complex, hence the nomenclature Activated Complex Theory (as well as ).

A further important aspect of invoking the activated complex is that for a reaction such as AB + C → A + BC, the path the reaction follows (if depicted on a three-dimensional ) is the minimum energy path. That is, the reaction will not follow along a reaction coor- dinate that requires any higher energetic cost than is minimally necessary. See Figure 36.19 in Engell and Reid for a represenation of a representative 3-D potential energy surface.

Assumptions of activated complex theory:

• Equilibrium exists between reactants and activated complex

5 • Reaction coordinate can be mapped onto a single energetic degree of freedom of the activated complex (i.e., a vibrational degree of freedom corresponding to bond-stretching)

The kinetic mechanism incorporating the activated complex is now:

k1 A + B )−−−*− AB† k−1

AB† −k→2 P roducts

The differential rate expression for A becomes:

d[A] = 0 = −k [A][B] + k [AB†] dt −1 −1

† † k1 Kc [AB ] = [A][B] = o [A][B] k−1 c Note the dependence of the on the standard state con- centration, co:

† [AB ] † o o AB c K† = c = c [A] [B] [A] [B] co co   The rate of product, P, formation (equal to the reaction rate) is:

d[P ] Rate = = k [AB†] dt 2 and since we know what [AB†] is from equilibrium of the reactants with the activated complex:

d[P ] k K† Rate = = 2 c [A][B] dt co

Now, we will consider how to refine our understanding of k2. Consider:

6 • We have formulated our discussion with the assumption that there is a single reaction coordinate mapped onto a vibrational degree of freedom

• Let this vibrational degree of freedom be a weak bond

• Rate of product formation = Rate of reaction = Frequency of vibration of weak bond (inverse seconds) (Maximum rate since we are taking every initial motion along that vibration coordinate to lead to product formation) (k2 = ν). • If every κ of the vibrational cycles leads to product formation, then k2 = κν

Thus, the reaction rate can be written as:

d[P ] κ ν K† Rate = = c [A][B] dt co Using elements of (the details of which are outside the current scope, but are covered in: Moore and Pearson, Kinetics and Mechanism, John Wiley, 1981, pp. 159-186)

κ k T K† k = B c 2 h co where the h in the denominator is Planck’s constant. Finally, if we recall that we can relate the equilibrium constant to the free energy change from reactant to activated complex,

† − † ∆G = RT ln(Kc )

Thus, the rate constant becomes (taking κ = 1, which is a good approx- imation for most situations (exceptions include surface reactions)):

k T † k = B e−∆G /RT 2 hco Since:

7 ∆G† = ∆H† − T ∆S†

we finally obtain the :

k T † † k = B e−∆H /RT e∆S /R 2 hco In the most general sense:

k T † † k(T ) = B e−∆H /RT e∆S /R hco We see that activated complex theory allows us to incorporate free energy as a metric for determining the rate constant. In this sense, entropic factors come into play as well. In the next section, we will discuss the relation between the parameters of Arrhenius theory and ∆S † and ∆H†.

4 Connection to Arrhenius Parameters

The quantities ∆H† and ∆S† are now related to the Arrhenius pre-exponential factor and activation energy, Ea.

Table 1. Relation Between Arrhenius and Eyring Parameters.

Phase / Molecularity Activation Energy, Ea Pre-Exponential, A † e kB T ∆S†/R Solution/Bimolecular Ea = ∆H + RT A = hco e

† e kB T ∆S†/R Solution/Unimolecular Ea = ∆H + RT A = h e

† e kB T ∆S†/R Gas/Unimolecular Ea = ∆H + RT A = h e

2 † e kB T ∆S†/R Gas/Bimolecular Ea = ∆H + 2RT A = hco e

3 † e kB T ∆S†/R Gas/Trimolecular Ea = ∆H + 3RT A = h(co)2 e

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